<p>Credit risk is the risk of losses due to the failure to fulfil the obliged payment from a debtor or a counterparty. It is one of the three major components of risks that a bank faces as defined in the new Basel Accord. The credit risk literature has experienced similar rapid growth as the credit market itself. There are currently four different approaches to analyzing credit risk: structural, reduced-form, incomplete information and hybrid models. Even though there are large volumes of published research papers and books on credit risk, our understanding and management skills in this area are still very limited as evidenced by the recent crash of the subprime market. This thesis combines three working papers on credit risk modeling and aims at adding some insights and contributions to the current credit risk literature.</p><p>In the first paper, we propose to randomize the initial condition of a generalized structural model, where the solvency ratio instead of the asset value is modeled explicitly. This initial randomization assumption is motivated by the fact that market players cannot observe the solvency ratio accurately. We find that positive short spreads can be produced due to imperfect observation on the risk factor. The two models we have considered, the Randomized Merton (RM)-II and the Randomized Black-Cox (RBC)-II, both have explicit expressions for Probability of Default (PD), Loss Given Default (LGD) and Credit Spreads (CS). In the RM-II model, both PD and LGD are found to be of order of √T, as the maturity T approaches zero. It therefore provides an example that has no well-defined default intensity but still admits positive short spreads. In the RBC-II model, the positive short spread is generated through the positive default intensity of the model. Because explicit formulas are available, these two Randomized Structure (RS) models are easily implemented and calibrated to the market data. This is illustrated by a calibration exercise on Ford Motor Corp. Credit Default Swap (CDS) spread data.</p> <p>In the second paper, we introduce the inverse-CIR (iCIR) intensity model of credit risk. A multi-firm intensity-based model is constructed where negative correlations are built through the negative correlation between the Cox-Ingersoll-Ross (CIR) process and its inverse. This parsimonious setting allows us to form rich correlation structures among short spreads of different firms, while keeping nonnegative conditions for interest rates and short spreads. The bond prices are given by explicit expressions involving confluent hypogeometric functions. This model can be regarded as an extension of the Ahn & Gao (1999) one factor iCIR model on interest rates to a multi-factor framework on credit risk.</p> <p> In the third paper, we derive several forms of the equity volatility as a function of the equity value, from the structural credit risk literature. We then propose a new jump to default model by taking the equity volatility to be of the form implied by the models of Leland (1994) and Leland & Toft (1996). This model involves a process we call the Dual-Jacobi process and which has explicit formulae for its moments. Gram-Charlier expansions are then applied to approximate bond and call prices. Our model generalizes Linetsky (2006) by incorporating a local volatility which is bounded below by a positive constant. This local volatility will decrease to a positive constant for increasing stock prices, making the stock process asymptotic to Geometric Brownian Motion (GBM). In this sence, our model is more realistic than Constant Elasticity of Variance (CEV) models.</p>